Requirements on a digital signal processing device to be used in a telephony network normally aim at minimizing the signal propagation delay imposed by the device, to reduce the total round trip delay which otherwise would affect the conversation quality negatively, see [1]. It is common for signal processing devices (such as noise reduction devices, echo cancellers, etc.) to apply some kind of linear filtering, serially connected to the signal path, to improve the speech signal. Two examples are:    1. Noise reduction devices, where the common solution is to apply filtering so that the disturbing background noise is reduced while at the same time retaining the speech so as to improve intelligibility.    2. Echo cancellation devices, where typically a linear adaptive filter (connected in parallel with the echo path) models the echo path to enable subtraction of a replica of the echo and thus a reduction of the echo level. However, due to imperfections in the adaptive algorithms and due to time varying and possibly non-linear echo paths, residual echo is left after the subtraction. This residual echo is typically removed by a non-linear processor (NLP), which detects the presence of the residual echo and, if found, removes it (optionally replacing it with some kind of generated comfort noise). The NLP is usually implemented as a switch that either passes or suppresses the signal applied to it. However, the NLP may also be implemented as a serially connected linear filter, as described in [2].
In both these telecommunication applications it is vital that the voice enhancement devices do not add to the round trip delay. Thus, it is desirable to minimize the delay of the signal processing filters. However, at the same time it is also desirable to maintain high stop-band rejection. Achieving high stop-band rejection is particularly important in echo cancellation applications, where a serially connected filter is required to suppress residual echo in certain frequency bands (the ones where residual echo dominates over true near end signal), while passing some other bands (the ones dominated by the true near end signal).
The derivative of the phase characteristic of a linear filter with respect to frequency is a measure of the time delay (i.e. group delay or envelope delay) that signal frequency components undergo in passing through the filter. A minimum-phase filter characteristic implies a minimum delay function, while a maximum-phase filter characteristic implies that the delay characteristic is also maximum. A mixed-phase system has a delay in between the two extremes. One solution to decrease the delay, which is imposed by a linear filter, is to convert it to a minimum-phase filter, which by definition has a minimal group delay through the filter.
Suppose that we have a finite impulse response (FIR) filter prototype H(ω) with real coefficients. The magnitude square value of its frequency response is
                                                                    H              ⁡                              (                ω                )                                                          2                =                                            H              ⁡                              (                z                )                                      ⁢                                          H                *                            (                              1                                  z                  *                                            )                                ⁢                      |                          z              =                              e                                  j                  ⁢                                                                          ⁢                  ω                                                                                        (        1        )            
Relationship (1) implies that if we replace a zero zk of the filter by its inverse 1/zk inside the unit circle, the magnitude characteristic of the filter does not change. Hence, a method for converting a finite impulse response (FIR) mixed/maximum phase filter to a minimum-phase filter is to mirror all zeros into the unit circle. However, this requires that the exact values of the zeros zk need to be calculated and would also require a large amount of divisions, which is inefficient in a typical real-time application on a digital signal processor (DSP). Mainly three workable other approaches exist for performing the conversion to a minimum-phase filter:    A. Spectral factorisation, e.g. using some approximating recursive algorithms as described in [3, 4]. However, a problem with spectral factorisation is that it is computationally demanding and that it needs an increasing number of iterations to reliably form a minimum-phase filter of a demanding filter prototype (i.e. a complicated desired magnitude response with high dynamic range) with high required stop band attenuation.    B. Employing homomorphic filtering as described in, for example, [5]. The method essentially determines the real cepstrum of the magnitude response of the desired digital filter, transforms this into a corresponding complex cepstrum representing a minimum-phase filter, and determines the minimum-phase digital filter corresponding to the complex cepstrum. The problem with this method is that it requires long discrete Fourier transforms (DFTs) to attain high stop band rejection in the resulting minimum-phase filter and thus becomes computationally demanding.    C. The errors imposed due to the usage of a discrete Fourier transform (DFT) in method B have been analysed in [6], and a method is proposed to reduce these errors. The method basically trusts on the usage of long DFTs together with truncation of the resulting impulse responses to minimise the effects of aliasing. This results in that the effective length of the impulse response will be half of the length of the used DFT. When using short DFTs, this leads to difficulties in obtaining the desired magnitude response of the prototype filter.